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A logarithmic function is a powerful mathematical tool used to determine the exponent required to reach a given number using a specified base. In simpler terms, for the expression \(\text{log}_{b}a\), we want to know the power required to raise the base \(b\) to get the value \(a\).
For instance, in \(\text{log}_{10}70\), we seek the power to which the base 10 must be raised to obtain 70. If you think about it, logarithms act like the reverse operation of exponentiation. Where exponentiation is about growing a number, the logarithm breaks it down to the exponent.
Understanding how to use logarithmic functions can simplify problems involving exponential relationships, making it a crucial concept in algebra and higher mathematics.

The base of a logarithm is essential for understanding its value. In logarithmic expressions like \(\text{log}_{b}a\), the base \(b\) is the number that gets raised to a certain power.
For instance, in the log expression \(\text{log}_{10}70\), the base here is 10. This means we're working with powers of 10.
To make sense of which power of 10 yields the number 70, consider some simple known values:

• \text{log}_{10}10 = 1\: 10 raised to the power of 1 equals 10.
• \text{log}_{10}100 = 2\: 10 raised to the power of 2 equals 100.

Matching 70, which falls between 10 and 100, suggests that \(\text{log}_{10}70\) is somewhere between 1 and 2. Just as described in the solution, the base number guides us to the correct exponent range.

Properties of logarithms help simplify complex calculations and solve log problems more efficiently. These properties include:

• Product Rule: \( \text{log}_{b}(xy) = \text{log}_{b}x + \text{log}_{b}y \)
• Quotient Rule: \( \text{log}_{b}(x/y) = \text{log}_{b}x - \text{log}_{b}y \)
• Power Rule: \( \text{log}_{b}(x^y) = y \text{log}_{b}x \)
• Change of Base Formula: \( \text{log}_{b}a = \frac{\text{log}_{c}a}{\text{log}_{c}b} \)

Using these properties, we can breakdown complex expressions into simpler parts.
In the given exercise, understanding that 70 falls between 10 and 100 allows us to conclude that \( \text{log}_{10}70 \) lies between 1 and 2.
Grasping these properties can transform seemingly complex logarithmic problems into manageable calculations.